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Model-Based Compressive Sensing
Richard G. Baraniuk, Volkan Cevher, Marco F. Duarte, Chinmay Hegde
Department of Electrical and Computer Engineering
Rice University
Abstract
Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquisition of sparse or
compressible signals that can be well approximated by justK ≪ N elements from anN -dimensional
basis. Instead of taking periodic samples, we measure innerproducts withM < N random vectors
and then recover the signal via a sparsity-seeking optimization or greedy algorithm. The standard CS
theory dictates that robust signal recovery is possible from M = O (K log(N/K)) measurements. The
goal of this paper is to demonstrate that it is possible to substantially decreaseM without sacrificing
robustness by leveraging more realistic signal models thatgo beyond simple sparsity and compressibility
by including dependencies between values and locations of the signal coefficients. We introduce a model-
based CS theory that parallels the conventional theory and provides concrete guidelines on how to create
model-based recovery algorithms with provable performance guarantees. A highlight is the introduction
of a new class of model-compressible signals along with a newsufficient condition for robust model-
compressible signal recovery that we dub the restricted amplification property (RAmP). The RAmP is
the natural counterpart to the restricted isometry property (RIP) of conventional CS. To take practical
advantage of the new theory, we integrate two relevant signal models — wavelet trees and block sparsity
— into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just
M = O (K) measurements. Extensive numerical simulations demonstrate the validity and applicability
of our new theory and algorithms.
Index Terms
Compressive sensing, sparsity, signal model, union of subspaces, wavelet tree, block sparsity
The authors are listed alphabetically. Email:richb, volkan, duarte, [email protected]; Web: dsp.rice.edu/cs. This work
was supported by the grants NSF CCF-0431150 and CCF-0728867, DARPA/ONR N66001-08-1-2065, ONR N00014-07-1-0936
and N00014-08-1-1112, AFOSR FA9550-07-1-0301, ARO MURI W311NF-07-1-0185, and the Texas Instruments Leadership
University Program.
I. INTRODUCTION
We are in the midst of a digital revolution that is enabling the development and deployment
of new sensors and sensing systems with ever increasing fidelity and resolution. The theoretical
foundation is the Shannon/Nyquist sampling theorem, whichstates that a signal’s information is
preserved if it is uniformly sampled at a rate at least two times faster than its Fourier bandwidth.
Unfortunately, in many important and emerging applications, the resulting Nyquist rate can be
so high that we end up with too many samples and must compress in order to store or transmit
them. In other applications the cost of signal acquisition is prohibitive, either because of a high
cost per sample, or because state-of-the-art samplers cannot achieve the high sampling rates
required by Shannon/Nyquist. Examples include radar imaging and exotic imaging modalities
outside visible wavelengths.
Transform compression systems reduce the effective dimensionality of anN-dimensional
signal x by re-representing it in terms of a sparse set of coefficientsα in a basis expansion
x = Ψα, with Ψ an N × N basis matrix. By sparse we mean that onlyK ≪ N of the
coefficientsα are nonzero and need to be stored or transmitted. By compressible we mean that
the coefficientsα, when sorted, decay rapidly enough to zero thatα can be well-approximated as
K-sparse. The sparsity and compressibility properties are pervasive in many signals of interest.
For example, smooth signals and images are compressible in the Fourier basis, while piecewise
smooth signals and images are compressible in a wavelet basis [1]; the JPEG and JPEG2000
standards are examples of practical transform compressionsystems based on these bases.
Compressive sensing(CS) provides an alternative to Shannon/Nyquist sampling when the
signal under acquisition is known to be sparse or compressible [2–4]. In CS, we measure
not periodic signal samples but rather inner products withM ≪ N measurement vectors. In
matrix notation, the measurementsy = Φx = ΦΨα, where the rows of theM × N matrix
Φ contain the measurement vectors. While the matrixΦΨ is rank deficient, and hence loses
information in general, it can be shown to preserve the information in sparse and compressible
signals if it satisfies the so-calledrestricted isometry property(RIP) [3]. Intriguingly, a large
2
class of random matrices have the RIP with high probability.To recover the signal from the
compressive measurementsy, we search for the sparsest coefficient vectorα that agrees with
the measurements. To date, research in CS has focused primarily on reducing both the number
of measurementsM (as a function ofN andK) and on increasing the robustness and reducing
the computational complexity of the recovery algorithm. Today’s state-of-the-art CS systems
can robustly recoverK-sparse and compressible signals from justM = O (K log(N/K)) noisy
measurements using polynomial-time optimization solversor greedy algorithms.
While this represents significant progress from Nyquist-rate sampling, our contention in this
paper is that it is possible to do even better by more fully leveraging concepts from state-of-the-
art signal compression and processing algorithms. In many such algorithms, the key ingredient is
a more realisticsignal modelthat goes beyond simple sparsity by codifying the inter-dependency
structureamong the signal coefficientsα.1 For instance, JPEG2000 and other modern wavelet
image coders exploit not only the fact that most of the wavelet coefficients of a natural image
are small but also the fact that the values and locations of the large coefficients have a particular
structure. Coding the coefficients according to a model for this structure enables these algorithms
to compress images close to the maximum amount possible – significantly better than a naıve
coder that just processes each large coefficient independently.
In this paper, we introduce a model-based CS theory that parallels the conventional theory and
provides concrete guidelines on how to create model-based recovery algorithms with provable
performance guarantees. By reducing the degrees of freedomof a sparse/compressible signal
by permitting only certain configurations of the large and zero/small coefficients, signal models
provide two immediate benefits to CS. First, they enable us toreduce, in some cases significantly,
the number of measurementsM required to stably recover a signal. Second, during signal
recovery, they enable us to better differentiate true signal information from recovery artifacts,
which leads to a more robust recovery.
1Obviously, sparsity and compressibility correspond to simple signal models where each coefficient is treated independently;
for example in a sparse model, the fact that the coefficientαi is large has no bearing on the size of anyαj , j 6= i. We will
reserve the use of the term “model” for situations where we are enforcing dependencies between the values and the locations
of the coefficientsαi.
3
To precisely quantify the benefits of model-based CS, we introduce and study several new
theoretical concepts that could be of more general interest. We begin with signal models forK-
sparse signals and make precise how the structure encoded ina signal model reduces the number
of potential sparse signal supports inα. Then using themodel-based restricted isometry property
(RIP) from [5, 6], we prove that suchmodel-sparse signalscan be robustly recovered from noisy
compressive measurements. Moreover, we quantify the required number of measurementsM
and show that for some modelsM is independent ofN . These results unify and generalize
the limited related work to date on signal models for strictly sparse signals [5–9]. We then
introduce the notion of amodel-compressible signal, whose coefficientsα are no longer strictly
sparse but have a structured power-law decay. To establish that model-compressible signals can
be robustly recovered from compressive measurements, we generalize the CS RIP to a new
restricted amplification property(RAmP). For some compressible signal models, the required
number of measurementsM is independent ofN .
To take practical advantage of this new theory, we demonstrate how to integrate signal
models into two state-of-the-art CS recovery algorithms, CoSaMP [10] and iterative hard thresh-
olding (IHT) [11]. The key modification is surprisingly simple: we merely replace the nonlinear
approximation step in these greedy algorithms with a model-based approximation. Thanks to our
new theory, both new model-based recovery algorithms have provable robustness guarantees for
both model-sparse and model-compressible signals.
To validate our theory and algorithms and demonstrate its general applicability and utility, we
present two specific instances of model-based CS and conducta range of simulation experiments.
The first model accounts for the fact that the large wavelet coefficients of piecewise smooth
signals and images tend to live on a rooted, connectedtree structure[12]. Using the fact that the
number of such trees is much smaller than(
NK
), the number ofK-sparse signal supports inN
dimensions, we prove that a tree-based CoSaMP algorithm needs onlyM = O (K) measurements
to robustly recover tree-sparse and tree-compressible signals. Figure 1 indicates the potential
performance gains on a tree-compressible, piecewise smooth signal.
The second model accounts for the fact that the large coefficients of many sparse signals clus-
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(a) test signal (b) CoSaMP (RMSE= 1.123)
(c) ℓ1-optimization (RMSE= 0.751) (d) model-based recovery (RMSE= 0.037)
Fig. 1. Example performance of model-based signal recovery. (a) Piecewise-smoothHeaviSinetest signal of length
N = 1024. This signal is compressible under a connected wavelet treemodel. Signal recovered fromM = 80 random
Gaussian measurements using (b) the iterative recovery algorithm CoSaMP, (c) standardℓ1 linear programming, and
(d) the wavelet tree-based CoSaMP algorithm from Section V.In all figures, root mean-squared error (RMSE) values
are normalized with respect to theℓ2 norm of the signal.
ter together [7, 8]. Such a so-calledblock sparsemodel is equivalent to ajoint sparsitymodel for
an ensemble ofJ , length-N signals [9], where the supports of the signals’ large coefficients are
shared across the ensemble. Using the fact that the number ofclustered supports is much smaller
than(
JNK
), we prove that a block-based CoSaMP algorithm needs onlyM = O
(K + K
Jlog(JN
K))
measurements to robustly recover block-sparse and block-compressible signals. Moreover, as the
number of signalsJ grows large, the number of measurements approachesM = O (K).
Our new theory and methods relate to a small body of previous work aimed at integrating
signal models with CS. Several groups have developed model-specific signal recovery algorithms
[5–8, 13–16]; however, their approach has either been ad hocor focused on a single model
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class. Previous work on unions of subspaces [5, 6, 17] has focused exclusively on strictly sparse
signals and has not considered feasible recovery algorithms. To the best of our knowledge, our
general framework for model-based recovery, the concept ofa model-compressible signal, and
the associated RAmP are new to the literature.
This paper is organized as follows. A review of the CS theory in Section II lays out the
foundational concepts that we extend to the model-based case in subsequent sections. Section III
develops the concept of model-sparse signals and introduces the concept of model-compressible
signals. We also quantify how signal models improve the measurement and recovery process
by exploiting the model-based RIP for model-sparse signalsand by introducing the RAmP for
model-compressible signals. Section IV indicates how to tune CoSaMP to incorporate model
information and establishes its robustness properties formodel-sparse and model-compressible
signals. Sections V and VI then specialize our theory to the special cases of wavelet tree and
block sparse signal models and report on a series of numerical experiments that validate our
theoretical claims. We conclude with a discussion in Section VII. To make the paper more
readable, all proofs are relegated to a series of appendices.
II. BACKGROUND ON COMPRESSIVESENSING
A. Sparse and compressible signals
Given a basisψiNi=1, we can represent every signalx ∈ RN in terms ofN coefficients
αiNi=1 asx =∑N
i=1 αiψi; stacking theψi as columns into theN ×N matrix Ψ, we can write
succinctly thatx = Ψα. In the sequel, we will assume without loss of generality that the signal
x is sparse or compressible in the canonical domain so that thesparsity basisΨ is the identity
andα = x.
A signalx is K-sparseif only K ≪ N entries ofx are nonzero. We call the set of indices
corresponding to the nonzero entries thesupportof x and denote it by supp(x). The set of all
K-sparse signals is the union of the(
NK
), K-dimensional subspaces aligned with the coordinate
axes inRN . We denote this union of subspaces byΣK .
Many natural and manmade signals are not strictly sparse, but can be approximated as such;
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we call such signalscompressible. Consider a signalx whose coefficients, when sorted in order
of decreasing magnitude, decay according to the power law
∣∣xI(i)
∣∣ ≤ S i−1/r, i = 1, . . . , N, (1)
whereI indexes the sorted coefficients. Thanks to the rapid decay oftheir coefficients, such
signals are well-approximated byK-sparse signals. LetxK ∈ ΣK represent the bestK-term
approximation ofx, which is obtained by keeping just the firstK terms in xI(i) from (1).
Denote the error of this approximation in theℓp norm as
σK(x)p := arg minx∈ΣK
‖x− x‖p = ‖x− xK‖p, (2)
where theℓp norm of the vectorx is defined as‖x‖p =(∑N
i=1 |xi|p)1/p
for 0 < p <∞. Then,
we have that
σK(x)p ≤ (rs)−1/p SK−s, (3)
with s = 1r− 1
p. That is, when measured in theℓp norm, the signal’s best approximation error
has a power-law decay with exponents asK increases. We dub such a signals-compressible.
The approximation of compressible signals by sparse signals is the basis oftransform coding
as is used in algorithms like JPEG and JPEG2000 [1]. In this framework, we acquire the full
N-sample signalx; compute the complete set of transform coefficientsα via α = Ψ−1x; locate
the K largest coefficients and discard the(N − K) smallest coefficients; and encode theK
values and locations of the largest coefficients. While a widely accepted standard, this sample-
then-compress framework suffers from three inherent inefficiencies: First, we must start with
a potentially large number of samplesN even if the ultimate desiredK is small. Second, the
encoder must compute all of theN transform coefficientsα, even though it will discard all
but K of them. Third, the encoder faces the overhead of encoding the locations of the large
coefficients.
B. Compressive measurements and the restricted isometry property (RIP)
Compressive sensing (CS) integrates the signal acquisition and compression steps into a
single process [2–4]. In CS we do not acquirex directly but rather acquireM < N linear
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measurementsy = Φx using anM×N measurement matrixΦ. We then recoverx by exploiting
its sparsity or compressibility. Our goal is to pushM as close as possible toK in order to
perform as much signal “compression” during acquisition aspossible.
In order to recover a good estimate ofx (the K largestxi’s, for example) from theM
compressive measurements, the measurement matrixΦ should satisfy therestricted isometry
property (RIP) [3].
Definition 1: An M × N matrix Φ has theK-restricted isometry property(K-RIP) with
constantδK if, for all x ∈ ΣK ,
(1− δK)‖x‖22 ≤ ‖Φx‖22 ≤ (1 + δK)‖x‖22. (4)
In words, theK-RIP ensures that all submatrices ofΦ of size M × K are close to an
isometry, and therefore distance (and information) preserving. Practical recovery algorithms
typically require thatΦ have a slightly stronger2K-RIP,3K-RIP, or higher-order RIP in order to
preserve distances betweenK-sparse vectors (which are2K-sparse in general), three-way sums
of K-sparse vectors (which are3K-sparse in general), and other higher-order structures.
While the design of a measurement matrixΦ satisfying theK-RIP is an NP-Complete
problem in general [3], random matrices whose entries are i.i.d. Gaussian, Bernoulli (±1), or
more generally subgaussian2 work with high probability providedM = O (K log(N/K)). These
random matrices also have a so-calleduniversalityproperty in that, for any choice of orthonormal
basis matrixΨ, ΦΨ has theK-RIP with high probability. This is useful when the signal issparse
not in the canonical domain but in basisΨ. A randomΦ corresponds to an intriguing data
acquisition protocol in which each measurementyj is a randomly weighted linear combination
of the entries ofx.
2A random variableX is called subgaussian if there existsc > 0 such thatE`
eXt´
≤ ec2t2/2 for all t ∈ R. Examples include
the Gaussian and Bernoulli random variables, as well as any bounded random variable. [18]
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C. Recovery algorithms
Since there are infinitely many signal coefficient vectorsx′ that produce the same set of
compressive measurementsy = Φx, to recover the “right” signal we exploit our a priori
knowledge of its sparsity or compressibility. For example,we could seek the sparsestx that
agrees with the measurementsy:
x = arg miny=Φx′
‖x′‖0, (5)
where theℓ0 “norm” of a vector counts its number of nonzero entries. While this optimization
can recover aK-sparse signal from justM = 2K compressive measurements, it is unfortunately
a combinatorial, NP-Complete problem; furthermore, the recovery is not stable in the presence
of noise.
Practical, stable recovery algorithms rely on the RIP (and therefore require at leastM =
O (K log(N/K)) measurements); they can be grouped into two camps. The first approach
convexifies theℓ0 optimization (5) to theℓ1 optimization
x = arg miny=Φx′
‖x′‖1. (6)
This corresponds to a linear program that can be solved in polynomial time [2, 3]. Adaptations to
deal with additive noise iny or x include basis pursuit with denoising (BPDN) [19], complexity-
based regularization [20], and the Dantzig Selector [21].
The second approach finds the sparsestx agreeing with the measurementsy through an
iterative, greedy search. Algorithms such as matching pursuit, orthogonal matching pursuit
[22], StOMP [23], iterative hard thresholding (IHT) [11], CoSaMP [10], and Subspace Pursuit
(SP) [24] all revolve around a bestL-term approximation for the estimated signal, withL varying
for each algorithm.
D. Performance bounds on signal recovery
GivenM = O (K log(N/K)) compressive measurements, a number of different CS signal
recovery algorithms, including all of theℓ1 techniques mentioned above and the CoSaMP, SP,
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and IHT iterative techniques, offer provably stable signalrecovery with performance close to
optimalK-term approximation (recall (3)). For a randomΦ, all results hold with high probability.
For a noise-free,K-sparse signal, these algorithms offer perfect recovery, meaning that the
signal x recovered from the compressive measurementsy = Φx is exactlyx = x.
For aK-sparse signalx whose measurements are corrupted by noisen of bounded norm
— that is, we measurey = Φx+ n — the mean-squared error of the recovered signalx is
‖x− x‖2 ≤ C‖n‖2, (7)
with C a small constant [2, 3, 10, 11].
For ans-compressible signalx whose measurements are corrupted by noisen of bounded
norm, the mean-squared error of the recovered signalx is
‖x− x‖2 ≤ C1‖x− xK‖2 + C21√K‖x− xK‖1 + C3‖n‖2. (8)
Using (3) we can simplify this expression to
‖x− x‖2 ≤C1SK
−s
√2s
+C2SK
−s
s− 1/2+ C3‖n‖2. (9)
III. B EYOND SPARSE AND COMPRESSIBLESIGNALS
While many natural and manmade signals and images can be described to first-order as sparse
or compressible, the support of their large coefficients often has an underlying inter-dependency
structure. This phenomenon has received only limited attention by the CS community to date [5–
8, 14–16]. In this section, we introduce a model-based theory of CS that captures such structure.
A model reduces the degrees of freedom of a sparse/compressible signal by permitting only
certain configurations of supports for the large coefficient. As we will show, this allows us to
reduce, in some cases significantly, the number of compressive measurementsM required to
stably recover a signal.
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A. Model-sparse signals
Recall from Section II-A that aK-sparse signal vectorx lives in ΣK ⊂ RN , which is a union
of(
NK
)subspaces of dimensionK. Other than itsK-sparsity, there are no further constraints
on the support or values of its coefficients. Asignal modelendows theK-sparse signalx with
additional structure that allows certainK-dimensional subspaces inΣK and disallows others [5,
6].
To state a formal definition of a signal model, letx|Ω represent the entries ofx corresponding
to the set of indicesΩ ⊆ 1, . . . , N, and letΩC denote the complement of the setΩ.
Definition 2: A signal modelMK is defined as the union ofmK canonicalK-dimensional
subspaces
MK =
mK⋃
m=1
Xm, such thatXm := x : x|Ωm ∈ RK , x|ΩC
m= 0,
where each subspaceXm contains all signalsx with supp(x) ∈ Ωm. Thus, the modelMK is
defined by the set of possible supportsΩ1, . . . ,ΩmK.
Signals fromMK are calledK-model sparse. Clearly,MK ⊆ ΣK and containsmK ≤(
NK
)
subspaces.
In Sections V and VI below we consider two concrete models forsparse signals. The first
model accounts for the fact that the large wavelet coefficients of piecewise smooth signals and
images tend to live on a rooted, connectedtree structure[12]. The second model accounts for
the fact that the large coefficients of sparse signals oftencluster together [7–9].
B. Model-based RIP
If we know that the signalx being acquired isK-model sparse, then we can relax the
RIP constraint on the CS measurement matrixΦ and still achieve stable recovery from the
compressive measurementsy = Φx [5, 6].
Definition 3: [5, 6] An M × N matrix Φ has theMK-restricted isometry property(MK-
RIP) with constantδMKif, for all x ∈MK , we have
(1− δMK)‖x‖22 ≤ ‖Φx‖22 ≤ (1 + δMK
)‖x‖22. (10)
11
To obtain a performance guarantee for model-based recoveryof K-model sparse signals in
additive measurement noise, we must define an enlarged unionof subspaces that includes sums
of elements in the model.
Definition 4: TheB-Minkowski sumfor the setMK , with B > 1 an integer, is defined as
MBK =
x =
B∑
r=1
x(r), with x(r) ∈MK
.
DefineMB(x,K) as the algorithm that obtains the best approximation ofx in the enlarged
union of subspacesMBK :
MB(x,K) = arg minx∈MB
K
‖x− x‖2.
We write M(x,K) := M1(x,K) when B = 1. Note that for many models, we will have
MBK ⊂MBK , and so the algorithmM(x,BK) will provide a strictly better approximation than
MB(x,K).
Our performance guarantee for model-sparse signal recovery will require that the measure-
ment matrixΦ be a near-isometry for all subspaces inMBK for someB > 1. This requirement
is a direct generalization of the2K-RIP, 3K-RIP, and higher-order RIPs from the conventional
CS theory.
Blumensath and Davies [5] have quantified the number of measurementsM necessary for
a random CS matrix to have theMK-RIP with a given probability.
Theorem 1: [5] LetMK be the union ofmK subspaces ofK-dimensions inRN . Then, for
any t > 0 and any
M ≥ 2
cδ2MK
(ln(2mK) +K ln
12
δMK
+ t
),
anM×N i.i.d. subgaussian random matrix has theMK-RIP with constantδMKwith probability
at least1− e−t.
This bound can be used to recover the conventional CS result by substitutingmK =(
NK
)≈
(Ne/K)K . TheMK-RIP property is sufficient for robust recovery of model-sparse signals, as
we show below in Section IV-B.
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C. Model-compressible signals
Just as compressible signals are “nearlyK-sparse” and thus live close to the union of
subspacesΣK in RN , model-compressible signals are “nearlyK-model sparse” and live close
to the restricted union of subspacesMK . In this section, we make this new concept rigorous.
Recall from (3) that we defined compressible signals in termsof the decay of theirK-term
approximation error.
The ℓ2 error incurred by approximatingx ∈ RN by the best model-based approximation in
MK is given by
σMK(x) := inf
x∈MK
‖x− x‖2 = ‖x−M(x,K)‖2.
The decay of this approximation error defines the model-compressibility of a signal.
Definition 5: The set ofs-model-compressible signalsis defined as
Ms =x ∈ R
N : σMK(x) ≤ SK−1/s, 1 ≤ K ≤ N, S <∞
.
Define |x|Ms as the smallest value ofS for which this condition holds forx ands.
We say thatx ∈Ms is ans-model-compressible signalunder the signal modelMK . These
approximation classes have been characterized for certainsignal models; see Section V for an
example.
D. Nested model approximations and residual subspaces
In conventional CS, the same requirement (RIP) is a sufficient condition for the stable
recovery of both sparse and compressible signals. In model-based recovery, however, the class
of compressible signals is much larger than that of sparse signals, since the set of subspaces
containing model-sparse signals does not span allK-dimensional subspaces. Therefore, we need
to introduce some additional tools to develop a sufficient condition for the stable recovery of
model-compressible signals.
We will pay particular attention to modelsMK that generatenested approximations, since
they are more amenable to analysis.
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Definition 6: A modelM = M1,M2, . . . has thenested approximation property(NAP)
if supp(M(x,K)) ⊂ supp(M(x,K ′)) for all K < K ′ and for allx ∈ RN .
In words, a model generates nested approximations if the support of the bestK ′-term model-
based approximation contains the support of the bestK-term model-based approximation for all
K < K ′. An important example of a NAP model is the standard compressible signal model of
(3).
When a model obeys the NAP, the support of the difference between the bestjK-term
model-based approximation and the best(j + 1)K-term model-based approximation of a signal
can be shown to lie in a small union of subspaces, thanks to thestructure enforced by the
model. This structure is captured by the set of subspaces that are included in each subsequent
approximation, as defined below.
Definition 7: The jth set of residual subspacesof sizeK is defined as
Rj,K(M) =u ∈ R
N such that u = M(x, jK)−M(x, (j − 1)K) for somex ∈ RN,
for j = 1, . . . , ⌈N/K⌉.
Under the NAP, each signalx in a model can be partitioned into its bestK-term
approximationxT1 , the additional components present in the best2K-term approximationxT2 ,
and so on, withx =∑⌈N/K⌉
j=1 xTjandxTj
∈ Rj,K(M) for eachj. Each signal partitionxTjis a
K-sparse signal, and thusRj,K(M) is a union of subspaces of dimensionK. We will denote
by Rj the number of subspaces that composeRj,K(M) and omit the dependence onM in the
sequel for brevity.
Intuitively, the norms of the partitions‖xTj‖2 decay asj increase for signals that are
compressible under the model. As the next subsection shows,this observation is instrumental in
relaxing the isometry restrictions on the measurement matrix Φ and bounding the recovery error
for s-model-compressible signals when the model obeys the NAP.
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E. The restricted amplification property (RAmP)
For exactlyK-model-sparse signals, we discussed in Section III-B that the number of
compressive measurementsM required for a random matrix to have theMK-RIP is determined
by the number of canonical subspacesmK via (11). Unfortunately, such model-sparse concepts
and results do not immediately extend to model-compressible signals. Thus, we develop a
generalization of theMK-RIP that we will use to quantify the stability of recovery for model-
compressible signals.
One way to analyze the robustness of compressible signal recovery in conventional CS is
to consider the tail of the signal outside itsK-term approximation as contributing additional
“noise” to the measurements of size‖Φ(x− xK)‖2 [10, 11, 25]. Consequently, the conventional
K-sparse recovery performance result can be applied with theaugmented noisen+ Φ(x−xK).
This technique can also be used to quantify the robustness ofmodel-compressible signal
recovery. The key quantity we must control is the amplification of the model-based approximation
residual throughΦ. The following property is a new generalization of the RIP and model-based
RIP.
Definition 8: A matrix Φ has the(ǫK , r)-restricted amplification property(RAmP) for the
residual subspacesRj,K of modelM if
‖Φu‖22 ≤ (1 + ǫK)j2r‖u‖22 (11)
for any u ∈ Rj,K for each1 ≤ j ≤ ⌈N/K⌉.
The regularity parameterr > 0 caps the growth rate of the amplification ofu ∈ Rj,K as a
function of j. Its value can be chosen so that the growth in amplification with j balances the
decay of the norm in each residual subspaceRj,K with j.
We can quantify the number of compressive measurementsM required for a random
measurement matrixΦ to have the RAmP with high probability; we prove the following in
Appendix I.
Theorem 2:Let Φ be anM × N matrix with i.i.d. subgaussian entries and let the set of
residual subspacesRj,K of modelM containRj subspaces of dimensionK for each1 ≤ j ≤
15
⌈N/K⌉. If
M ≥ max1≤j≤⌈N/K⌉
1(jr√
1 + ǫK − 1)2(
2K + 4 lnRjN
K+ 2t
), (12)
then the matrixΦ has the(ǫK , r)-RAmP with probability1− e−t.
The order of the bound of Theorem 2 is lower thanO (K log(N/K)) as long as the number
of subspacesRj grows slower thanNK .
Armed with the RaMP, we can state the following result, whichwill provide robustness for
the recovery of model-compressible signals; see Appendix II for the proof.
Theorem 3:Let x ∈ Ms be ans-model compressible signal under a modelM that obeys
the NAP. If Φ has the(ǫK , r)-RAmP andr = s− 1, then we have
‖Φ(x−M(x,K))‖2 ≤√
1 + ǫKK−s ln
⌈N
K
⌉|x|Ms .
IV. M ODEL-BASED SIGNAL RECOVERY ALGORITHMS
To take practical advantage of our new theory for model-based CS, we demonstrate how to
integrate signal models into two state-of-the-art CS recovery algorithms, CoSaMP [10] (in this
section) and iterative hard thresholding (IHT) [11] (in Appendix III). The key modification is
simple: we merely replace the bestK-term approximation step in these greedy algorithms with a
bestK-term model-based approximation. Since at each iteration we need only search over themK
subspaces ofMK rather than(
NK
)subspaces ofΣK , fewer measurements will be required for the
same degree of robust signal recovery. Or, alternatively, using the same number of measurements,
more accurate recovery can be achieved. After presenting the modified CoSaMP algorithm, we
prove robustness guarantees for both model-sparse and model-compressible signals.
A. Model-based CoSaMP
We choose to modify the CoSaMP algorithm [10] for two reasons. First, it has robust
recovery guarantees that are on par with the best convex optimization-based approaches. Second,
16
Algorithm 1 Model-based CoSaMPInputs: CS matrixΦ, measurementsy, modelMK
Output:K-sparse approximationx to true signalx
x0 = 0, r = y, i = 0 initializewhile halting criterion falsedo
i← i+ 1
e← ΦT r form signal residual estimateΩ← supp(M2(e,K)) prune signal residual estimate according to signal modelT ← Ω ∪ supp(xi−1) merge supportsb|T ← Φ†
T y, b|T C ← 0 form signal estimatexi ←M(b,K) prune signal estimate according to signal modelr ← y − Φxi update measurement residual
end while
return x← xi
it has a simple iterative, greedy structure based on a bestBK-term approximation (withB a
small integer) that is easily modified to incorporate a bestBK-term model-based approximation
MB(K, x). Pseudocode for the modified algorithm is given in Algorithm1.
We now study the performance of model-based CoSaMP signal recovery on model-sparse
signals and model-compressible signals.
B. Performance of model-sparse signal recovery
A robustness guarantee for noisy measurements of model-sparse signals can be obtained
using the model-based RIP (10). The following theorem is proven in Appendix IV.
Theorem 4:Let x ∈MK and lety = Φx+ n be a set of noisy CS measurements. IfΦ has
anM4K-RIP constant ofδM4
K≤ 0.1, then the signal estimatexi obtained from iterationi of the
model-based CoSaMP algorithm satisfies
‖x− xi‖2 ≤ 2−i‖x‖2 + 15‖n‖2. (13)
17
C. Performance of model-compressible signal recovery
Using the new tools introduced in Section III, we can providea robustness guarantee for
noisy measurements of model-compressible signals, using the RAmP as a condition on the
measurement matrixΦ.
Theorem 5:Let x ∈ Ms be ans-model-compressible signal from a modelM that obeys
the NAP, and lety = Φx + n be a set of noisy CS measurements. IfΦ has theM4K-RIP with
δM4K≤ 0.1 and the(ǫK , r)-RAmP with ǫK ≤ 0.1 and r = s − 1, then the signal estimatexi
obtained from iterationi of the model-based CoSaMP algorithm satisfies
‖x− xi‖2 ≤ 2−i‖x‖2 + 35(‖n‖2 + |x|MsK
−s(1 + ln⌈N/K⌉)). (14)
To prove the theorem, we first bound the recovery error for ans-model-compressible signal
x ∈ Ms when the matrixΦ has the(ǫK , r)-RAmP with r ≤ s − 1. Then, using Theorems 3
and 4, we can easily prove the result by following the analogous proof in [10].
D. Robustness to model mismatch
We now analyze the robustness of model-based CS recovery tomodel mismatch, which occurs
when the signal being recovered from compressive measurements does not conform exactly to
the model used in the recovery algorithm.
We begin with optimistic results for signals that are “close” to matching the recovery model.
First consider a signalx that is notK-model sparse as the recovery algorithm assumes but rather
(K + κ)-model sparse for some small integerκ. This signal can be decomposed intoxK , the
signal’sK-term model-based approximation, andx − xK , the error of this approximation. For
κ ≤ K, we have thatx−xK ∈ R2,K . If the matrixΦ has the(ǫK , r)-RAmP, then it follows than
‖Φ(x− xK)‖2 ≤ 2r√
1 + ǫK‖x− xK‖2. (15)
18
Using equations (13) and (15), we obtain the following guarantee for theith iteration of model-
based CoSaMP:
‖x− xi‖2 ≤ 2−i‖x‖2 + 16 · 2r√
1 + ǫK‖x− xK‖2 + 15‖n‖2.
By noting that‖x− xK‖2 is small, we obtain a guarantee that is close to (13).
Second, consider a signalx that is nots-model compressible as the recovery algorithm
assumes but rather(s− ǫ)-model compressible. The following bound can be obtained under the
conditions of Theorem 5 by modifying the argument in Appendix II:
‖x− xi‖2 ≤ 2−i‖x‖2 + 35
(‖n‖2 + |x|MsK
−s
(1 +⌈N/K⌉ǫ − 1
ǫ
)).
As ǫ becomes smaller, the factor⌈N/K⌉ǫ−1ǫ
approacheslog⌈N/K⌉, matching (14). In summary,
as long as the deviations from the model-sparse and model-compressible models are small, our
model-based recovery guarantees still apply within a smallbounded constant factor.
We end with a more pessimistic, worst-case result for signals that are arbitrarily far away
from model-sparse or model-compressible. Consider such anarbitraryx ∈ RN and compute its
nested model-based approximationsxjK = M(x, jK), j = 1, . . . , ⌈N/K⌉. If x is not model-
compressible, then the model-based approximation errorσjK(x) is not guaranteed to decay asj
decreases. Additionally, the number of residual subspacesRj,K could be as large as(
NK
); that is,
the jth difference between subsequent model-based approximations xTj= xjK − x(j−1)K might
lie in any arbitraryK-dimensional subspace. This worst case is equivalent to setting r = 0 and
Rj =(
NK
)in Theorem 2. It is easy to see that this condition on the number of measurements
M is nothing but the standard RIP for CS. Hence, if inflate the number of measurements to
M = O (K log(N/K)) (the usual number for conventional CS), the performance of model-based
CoSaMP recovery on an arbitrary signalx follows theK-term model-basedapproximation ofx
within a bounded constant factor.
E. Computational complexity of model-based recovery
The computational complexity of a model-based signal recovery algorithm differs from
that of a standard algorithm by two factors. The first factor is the reduction in the number
19
of measurementsM necessary for recovery: since most current recovery algorithms have a
computational complexity that is linear in the number of measurements, any reduction inM
reduces the total complexity. The second factor is the cost of the model-based approximation.
TheK-term approximation used in most current recovery algorithms can be implemented with a
simple sorting operation (O (N logN) complexity, in general). Ideally, the signal model should
support a similarly efficient approximation algorithm.
To validate our theory and algorithms and demonstrate theirgeneral applicability and utility,
we now present two specific instances of model-based CS and conduct a range of simulation
experiments.
V. EXAMPLE : WAVELET TREE MODEL
Wavelet decompositions have found wide application in the analysis, processing, and
compression of smooth and piecewise smooth signals becausethese signals areK-sparse and
compressible, respectively [1]. Moreover, the wavelet coefficients can be naturally organized
into a tree structure, and for many kinds of natural and manmade signals the largest coefficients
cluster along the branches of this tree. This motivates a connected tree model for the wavelet
coefficients [26–28].
While CS recovery for wavelet-sparse signals has been considered previously [14–16],
the resulting algorithms integrated the tree constraint inan ad-hoc fashion. Furthermore, the
algorithms provide no recovery guarantees or bounds on the necessary number of compressive
measurements.
A. Tree-sparse signals
We first describe tree sparsity in the context of sparse wavelet decompositions. We focus
on one-dimensional signals and binary wavelet trees, but all of our results extend directly to
d-dimensional signals and2d-ary wavelet trees.
Consider a signalx of lengthN = 2I , for an integer value ofI. The wavelet representation
20
...Fig. 2. Binary wavelet tree for a one-dimensional signal. The squares denote the large wavelet coefficients that arise
from the discontinuities in the piecewise smooth signal drawn below; the support of the large coefficients forms a
rooted, connected tree.
of x is given by
x = v0ν +
I−1∑
i=0
2i−1∑
j=0
wi,jψi,j ,
where ν is the scaling function andψi,j is the wavelet function at scalei and offsetj. The
wavelet transform consists of the scaling coefficientv0 and wavelet coefficientswi,j at scalei,
0 ≤ i ≤ I − 1, and positionj, 0 ≤ j ≤ 2i − 1. In terms of our earlier matrix notation,x has
the representationx = Ψα, whereΨ is a matrix containing the scaling and wavelet functions as
columns, andα = [v0 w0,0 w1,0 w1,1 w2,0 . . .]T is the vector of scaling and wavelet coefficients.
We are, of course, interested in sparse and compressibleα.
The nested supports of the wavelets at different scales create a parent/child relationship
between wavelet coefficients at different scales. We say that wi−1,⌊j/2⌋ is the parent of wi,j
and thatwi+1,2j and wi+1,2j+1 are thechildren of wi,j. These relationships can be expressed
graphically by the wavelet coefficient tree in Figure 2.
Wavelet functions act as local discontinuity detectors, and using the nested support property
of wavelets at different scales, it is straightforward to see that a signal discontinuity will give
rise to a chain of large wavelet coefficients along a branch ofthe wavelet tree from a leaf to
the root. Moreover, smooth signal regions will give rise to regions of small wavelet coefficients.
This “connected tree” property has been well-exploited in anumber of wavelet-based processing
21
[12, 29, 30] and compression [31, 32] algorithms. In this section, we will specialize the theory
developed in Sections III and IV to a connected tree modelT .
A set of wavelet coefficientsΩ forms aconnected subtreeif, whenever a coefficientwi,j ∈ Ω,
then its parentwi−1,⌊j/2⌋ ∈ Ω as well. Each such setΩ defines a subspace of signals whose support
is contained inΩ; that is, all wavelet coefficients outsideΩ are zero. In this way, we define the
modelTK as the union of allK-dimensional subspaces corresponding to supportsΩ that form
connected subtrees.
Definition 9: Define the set ofK-tree sparse signalsas
TK =
x = v0ν +
I−1∑
i=0
2i∑
j=1
wi,jψi,j : w|ΩC = 0, |Ω| = K,Ω forms a connected subtree
.
To quantify the number of subspaces inTK , it suffices to count the number of distinct
connected subtrees of sizeK in a binary tree of sizeN . We prove the following result in
Appendix V.
Proposition 1: The number of subspaces inTK obeysTK ≤ 4K+4
Ke2 for K ≥ log2N and
TK ≤ (2e)K
K+1for K < log2N .
B. Tree-based approximation
To implement tree-based signal recovery, we seek an efficient algorithm T(x,K) to solve
the optimal approximation
xTK = arg minx∈TK
‖x− x‖2. (16)
Fortuitously, an efficient solver exists, called thecondensing sort and select algorithm(CSSA)
[26–28]. Recall that subtree approximation coincides withstandardK-term approximation (and
hence can be solved by simply sorting the wavelet coefficients) when the wavelet coefficients
are monotonically nonincreasing along the tree branches out from the root. The CSSA solves
(16) in the case of general wavelet coefficient values bycondensingthe nonmonotonic segments
of the tree branches using an iterative sort-and-average routine. The condensed nodes are called
“supernodes”. Condensing a large coefficient far down the tree accounts for the potentially large
cost (in terms of the total budget of tree nodesK) of growing the tree to that point.
22
The CSSA can also be interpreted as a greedy search among the nodes. For each node in the
tree, the algorithm calculates the average wavelet coefficient magnitude for each subtree rooted
at that node, and records the largest average among all the subtrees as the energy for that node.
The CSSA then searches for the unselected node with the largest energy and adds the subtree
corresponding to the node’s energy to the estimated supportas a supernode [28].
Since the first step of the CSSA involves sorting all of the wavelet coefficients, overall it
requiresO (N logN) computations. However, once the CSSA grows the optimal treeof sizeK,
it is trivial to determine the optimal trees of size< K and computationally efficient to grow the
optimal trees of size> K [26].
The constrained optimization (16) can be rewritten as an unconstrained problem by
introducing the Lagrange multiplierλ [33]:
minx∈T‖x− x‖22 + λ(‖α‖0 −K),
where T = ∪Nn=1Tn and α are the wavelet coefficients ofx. Except for the inconsequential
λK term, this optimization coincides with Donoho’scomplexity penalized sum of squares[33],
which can be solved in onlyO (N) computations using coarse-to-fine dynamic programming on
the tree. Its primary shortcoming is the nonobvious relationship between the tuning parameter
λ and and the resulting sizeK of the optimal connected subtree.
C. Tree-compressible signals
Specializing Definition 2 from Section III-C toT , we make the following definition.
Definition 10: Define the set ofs-tree compressible signalsas
Ts =x ∈ R
N : ‖x− T(x,K)‖2 ≤ SK−s, 1 ≤ K ≤ N, S <∞.
Furthermore, define|x|Ts as the smallest value ofS for which this condition holds forx ands.
Tree approximation classes contain signals whose wavelet coefficients have a loose (and
possibly interrupted) decay from coarse to fine scales. These classes have been well-characterized
for wavelet-sparse signals [27, 28, 32] and are intrinsically linked with the Besov spaces
23
Bsq(Lp([0, 1])). Besov spaces contain functions of one or more continuous variables that have
(roughly speaking)s derivatives inLp([0, 1]); the parameterq provides finer distinctions of
smoothness. When a Besov space signalxa with s > 1/p − 1/2 is sampled uniformly and
converted to a length-N vectorx, its wavelet coefficients belong to the tree approximation space
Ts, with
|xN |Ts ≍ ‖xa‖Lp([0,1]) + ‖xa‖Bsq (Lp([0,1])),
where “≍” denotes an equivalent norm. The same result holds ifs = 1/p− 1/2 andq ≤ p.
D. Stable tree-based recovery from compressive measurements
For tree-sparse signals, by applying Theorem 1 and Proposition 1, we find that a subgaussian
random matrix has theTK-RIP property with constantδTKand probability1−e−t if the number
of measurements obeys
M ≥
2cδ2
TK
(K ln 48
δTK+ ln 512
Ke2 + t)
if K < log2N,
2cδ2
TK
(K ln 24e
δTK
+ ln 2K+1
+ t)
if K ≥ log2N,
Thus, the number of measurements necessary for stable recovery of tree-sparse signals is linear
in K, without the dependence onN present in conventional non-model-based CS recovery.
For tree-compressible signals, we must quantify the numberof subspacesRj in each residual
set Rj,K for the approximation class. We can then apply the theory of Section IV-C with
Proposition 1 to calculate smallest allowableM via Theorem 5.
Proposition 2: The number ofK-dimensional subspaces that compriseRj,K obeys
Rj ≤
(2e)K(2j+1)
(Kj+K+1)(Kj+1)if 1 ≤ j <
⌊log2 N
K
⌋,
2(3j+2)K+8ejK
(Kj+1)K(j+1)e2 if j =⌊
log2 NK
⌋,
4(2j+1)K+8
K2j(j+1)e4 if j >⌊
log2 NK
⌋.
(17)
Using Proposition 2 and Theorem 5, we obtain the following condition for the matrixΦ to
have the RAmP, which is proved in Appendix VI.
24
Proposition 3: Let Φ be anM ×N matrix with i.i.d. subgaussian entries.. If
M ≥
2
(√
1+ǫK−1)2
(10K + 2 ln N
K(K+1)(2K+1)+ t)
if K ≤ log2N,
2
(√
1+ǫK−1)2
(10K + 2 ln 601N
K3 + t)
if K > log2N,
then the matrixΦ has the(ǫK , s)-RAmP for modelT and alls > 0.5 with probability1− e−t.
Both cases give a simplified bound on the number of measurements required asM = O (K),
which is a substantial improvement over theM = O (K log(N/K)) required by conventional
CS recovery methods. Thus, whenΦ satisfies Proposition 3, we have the guarantee (14) for
sampled Besov space signals fromBsq(Lp([0, 1])).
E. Experiments
We now present the results of a number of numerical experiments that illustrate the
effectiveness of a tree-based recovery algorithm. Our consistent observation is that explicit
incorporation of the model in the recovery process significantly improves the quality of recovery
for a given number of measurements. In addition, model-based recovery remains stable when the
inputs are no longer tree-sparse, but rather are tree-compressible and/or corrupted with differing
levels of noise. We employ the model-based CoSaMP recovery of Algorithm 1 with a CSSA-
based approximation step in all experiments.
We first study one-dimensional signals that match the connected wavelet-tree model described
above. Among such signals is the class of piecewise smooth functions, which are commonly
encountered in analysis and practice.
Figure 1 illustrates the results of recovering the tree-compressibleHeaviSinesignal of length
N = 1024 from M = 80 noise-free random Gaussian measurements using CoSaMP,ℓ1-norm
minimization using thel1 eq solver from theℓ1-Magic toolbox,3 and our tree-based recovery
algorithm. It is clear that the number of measurements (M = 80) is far fewer than the minimum
number required by CoSaMP andℓ1-norm minimization to accurately recover the signal. In
contrast, tree-based recovery usingK = 26 is accurate and uses fewer iterations to converge
3http://www.acm.caltech.edu/l1magic.
25
2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
M/KA
vera
ge n
orm
aliz
ed e
rror
mag
nitu
de Model−based recoveryCoSaMP
Fig. 3. Performance of CoSaMP vs. wavelet tree-based recovery on a class of piecewise-cubic signals as a function
of M/K.
than conventional CoSaMP. Moreover, the normalized magnitude of the squared error for tree-
based recovery is equal to 0.037, which is remarkably close to the error between the noise-free
signal and itsbestK-term tree-approximation (0.036).
Figure 3 illustrates the results of a Monte Carlo simulationstudy on the impact of the number
of measurementsM on the performance of model-based and conventional recovery for a class
of tree-sparse piecewise-polynomial signals. Each data point was obtained by measuring the
normalized recovery error of 500 sample trials. Each sampletrial was conducted by generating
a new piecewise-polynomial signal with five polynomial pieces of cubic degree and randomly
placed discontinuities, computing its bestK-term tree-approximation using the CSSA, and then
measuring the resulting signal using a matrix with i.i.d. Gaussian entries. Model-based recovery
attains near-perfect recovery atM = 3K measurements, while CoSaMP only matches this
performance atM = 5K. We defer a full Monte Carlo comparison of our method with the
much more computationally demandingℓ1-norm minimization to future work. In practice, we
have noticed that CoSaMP andℓ1-norm minimization offer similar recovery trends; consequently,
we can expect that model-based recovery will offer a similardegree of improvement overℓ1-norm
minimization.
Further, we demonstrate that model-based recovery performs stably in the presence of
26
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1CoSaMP (M = 5K)
))
Max
imum
nor
mal
ized
rec
over
y er
ror
Fig. 4. Robustness to measurement noise for standard and wavelet tree-based CS recovery algorithms. We plot the
maximum normalized recovery error over 200 sample trials asa function of the expected signal-to-noise ratio. The
linear growth demonstrates that model-based recovery possesses the same robustness to noise as CoSaMP andℓ1-norm
minimization.
measurement noise. We generated sample piecewise-polynomial signals as above, computed
their bestK-term tree-approximations, computedM measurements of each approximation, and
finally added Gaussian noise of expected norm‖n‖2 to each measurement. Then, we recovered
the signal using CoSaMP and model-based recovery and measured the recovery error in each case.
For comparison purposes, we also tested the recovery performance of aℓ1-norm minimization
algorithm that accounts for the presence of noise, which hasbeen implemented as thel1 qc
solver in theℓ1-Magic toolbox. First, we determined the lowest value ofM for which the
respective algorithms provided near-perfect recovery in the absence of noise in the measurements.
This corresponds toM = 3.5K for model-based recovery,M = 5K for CoSaMP, andM = 4.5K
for ℓ1 minimization. Next, we generated 200 sample tree-modeled signals, computedM noisy
measurements, recovered the signal using the given algorithm and recorded the recovery error.
Figure 4 illustrates the growth in maximum normalized recovery error (over the 200 sample
trials) as a function of the expected measurement signal-to-noise ratio for the tree algorithms. We
observe similar stability curves for all three algorithms,while noting that model-based recovery
offers this kind of stability using significantly fewer measurements.
27
(a) Peppers (b) CoSaMP (c) model-based recovery
(RMSE = 22.8) (RMSE = 11.1)
Fig. 5. Example performance of standard and model-based recovery on images. (a)N = 128× 128 = 16384-pixel
Pepperstest image. Image recovery fromM = 5000 compressive measurements using (b) conventional CoSaMP and
(c) our wavelet tree-based algorithm.
Finally, we turn to two-dimensional images and a wavelet quadtree model. The connected
wavelet-tree model has proven useful for compressing natural images [27]; thus, our algorithm
provides a simple and provably efficient method for recovering a wide variety of natural images
from compressive measurements. An example of recovery performance is given in Figure 5. The
test image (Peppers) is of sizeN = 128 × 128 = 16384 pixels, and we computedM = 5000
random Gaussian measurements. Model-based recovery againoffers higher performance than
standard signal recovery algorithms like CoSaMP, both in terms of recovery mean-squared error
and visual quality.
VI. EXAMPLE : BLOCK-SPARSE SIGNALS AND SIGNAL ENSEMBLES
In a block-sparsesignal, the locations of the significant coefficients cluster in blocks under
a specific sorting order. Block-sparse signals have been previously studied in CS applications,
including DNA microarrays and magnetoencephalography [7,8]. An equivalent problem arises
in CS for signal ensembles, such as sensor networks and MIMO communication [8, 9, 34]. In this
case, several signals share a common coefficient support set. For example, when a frequency-
sparse acoustic signal is recorded by an array of microphones, then all of the recorded signals
28
contain the same Fourier frequencies but with different amplitudes and delays. Such a signal
ensemble can be re-shaped as a single vector by concatenation, and then the coefficients can be
rearranged so that the concatenated vector exhibits block sparsity.
It has been shown that the block-sparse structure enables signal recovery from a reduced
number of CS measurements, both for the single signal case [7, 8] and the signal ensemble
case [9], through the use of specially tailored recovery algorithm [7, 8, 35]. However, the
robustness guarantees for such algorithms either are restricted to exactly sparse signals and
noiseless measurements, do not have explicit bounds on the number of necessary measurements,
or are asymptotic in nature.
In this section, we formulate the block sparsity signal model as a union of subspaces and
pose an approximation algorithm on this union of subspaces.The approximation algorithm is
used to implement block-based signal recovery. We also define the corresponding class of block-
compressible signals and quantify the number of measurements necessary for robust recovery.
A. Block-sparse signals
Consider a classS of signal vectorsx ∈ RJN , with J andN integers. This signal can be
reshapped into aJ × N matrix X, and we use both notations interchangeably in the sequel.
We will restrict entire columns ofX to be part of the support of the signal as a group. That
is, signalsX in a block-sparse model have entire columns as zeros or nonzeros. The measure
of sparsity forX is its number of nonzero columns. More formally, we make the following
definition.
Definition 11: [7, 8] Define the set ofK-block sparse signalsas
SK = X = [x1 . . . xN ] ∈ RJ×N such thatxn = 0 for n /∈ Ω,Ω ⊆ 1, . . . , N, |Ω| = K.
It is important to note that aK-block sparse signal has sparsityKJ , which is dependent
on the size of the blockJ . We can extend this formulation to ensembles ofJ , length-N signals
with common support. Denote this signal ensemble byx1, . . . , xJ, with xj ∈ RN , 1 ≤ j ≤ J .
We formulate a matrix representationX of the ensemble that features the signalxj in its jth
row: X = [x1 . . . xN ]T . The matrixX features the same structure as the matrixX obtained
29
from a block-sparse signal; thus, the matrixX can be converted into a block-sparse vectorx
that represents the signal ensemble.
B. Block-based approximation
To pose a the block-based approximation algorithm, we need to define the mixed norm of
a matrix.
Definition 12: The (p, q) mixed normof the matrixX = [x1 x2 . . . xN ] is defined as
‖X‖(p,q) =
(N∑
n=1
‖xn‖qp
)1/q
.
Whenq = 0, ‖X‖(p,0) simply counts the number of nonzero columns inX.
We immediately find that‖X‖(p,p) = ‖x‖p, with x the vectorization ofX. Intuitively, we
pose the algorithmS(X,K) to obtain the best block-based approximation of the signalX as
follows:
XSK = arg min
X∈RJ×N‖X − X‖(2,2) subject to‖X‖(2,0) ≤ K. (18)
It is easy to show that to obtain the approximation, it suffices to perform column-wise hard
thresholding: letρ be theK th largestℓ2-norm among the columns ofX. Then our approximation
algorithm isS(X,K) = XSK = [xSK,1 . . . xSK,N ], where
xSK,n =
xn ‖xn‖2 ≥ ρ,
0 ‖xn‖2 < ρ,
for each1 ≤ j ≤ J and1 ≤ n ≤ N . Alternatively, a recursive approximation algorithm can be
obtained by sorting the columns ofX by their ℓ2 norms, and then selecting the largest columns.
The complexity of this sorting process isO (NJ +N logN).
C. Block-compressible signals
The approximation class under the block-compressible model corresponds to signals with
blocks whoseℓ2 norm has a power-law decay rate.
30
Definition 13: We define the set ofs-block compressible signals as
Ss = X = [x1 . . . xN ] ∈ RJ×N s.t. ‖xI(i)‖2 ≤ Si−s−1/2, 1 ≤ i ≤ N, S <∞,
whereI indexes the sorted column norms.
We say thatX is an s-block compressible signal ifX ∈ Ss. For such signals, we have‖X −XK‖(2,2) = σSK
(x) ≤ S1K−s, and‖X−XK‖(2,1) ≤ S2K
1/2−s. Note that the block-compressible
signal model does not impart a structure to the decay of the signal coefficients, so that the sets
Rj,K are equal for all values ofj; due to this property, the(δSK, s)-RAmP is implied by the
SK-RIP. Taking this into account, we can derive the following result from [10], which is proven
similarly to Theorem 4.
Theorem 6:Let x be a signal from modelS, and lety = Φx + n be a set of noisy CS
measurements. IfΦ has theS4K-RIP with δS4
K≤ 0.1, then the estimate obtained from iterationi
of block-based CoSaMP, using the approximation algorithm (18), satisfies
‖x− xi‖2 ≤ 2−i‖x‖2 + 20
(‖X −XS
K‖(2,2) +1√K‖X −XS
K‖(2,1) + ‖n‖2).
Thus, the algorithm provides a recovered signal of similar quality to approximations ofX
by a small number of nonzero columns. When the signalx is K-block sparse, we have that
||X − XSK‖(2,2) = ||X − XS
K‖(2,1) = 0, obtaining the same result as Theorem 4, save for a
constant factor.
D. Stable block-based recovery from compressive measurements
Since Theorem 6 poses the same requirement on the measurement matrix Φ for sparse and
compressible signals, the same number of measurementsM is required to provide performance
guarantees for block-sparse and block-compressible signals. The classSK containsS =(
NK
)
subspaces of dimensionJK. Thus, a subgaussian random matrix has theSK-RIP property with
constantδSKand probability1− e−t if the number of measurements obeys
M ≥ 2
cδ2SK
(K
(ln
2N
K+ J ln
12
δSK
)+ t
). (19)
31
The first term in this bound matches the order of the bound for conventional CS, while the
second term introduces a linear dependence on the size of theblock J . This shows that the
number of measurements required for robust recovery scalesasM = O (KJ +K log(N/K)),
which is a substantial improvement over theM = O (JK log(N/K)) that would be required by
conventional CS recovery methods. When the size of the blockJ is larger thanlog(N/K), then
this term becomesO (KJ); that is, it is linear on the total sparsity of the block-sparse signal.
We note in passing that the bound on the number of measurements (19) assumes a dense
subgaussian measurement matrix, while the measurement matrices used in [9] have a block-
diagonal. structure. To obtain measurements from anM × JN dense matrix in a distributed
setting, it suffices to partition the matrix intoJ pieces of sizeM × N and calculate the CS
measurements at each sensor with a corresponding matrix; these individual measurements are
then summed to obtain the complete measurement vector. For large J , (19) implies that the
total number of measurements required for recovery of the signal ensemble is lower than the
bound for the case where each signal recovery is performed independently for each signal (M
= O (JK log(N/K))).
E. Experiments
We conducted several numerical experiments comparing model-based recovery to CoSaMP
in the context of block-sparse signals. We employ the model-based CoSaMP recovery of
Algorithm 1 with the block-based approximation algorithm (18) in all cases. For brevity, we
exclude a thorough comparison of our model-based algorithmwith ℓ1-based optimization and
defer it to future work. In practice, we observed that our algorithm performs several times faster
than convex optimization-based procedures.
Figure 6 illustrates anN = 4096 signal that exhibits block sparsity, and its recovered version
using CoSaMP and model-based recovery. The block sizeJ = 64 and there wereK = 6 active
blocks in the signal. We observe the clear advantage of usingthe block-sparsity model in signal
recovery.
We now consider block-compressible signals. An example recovery is illustrated in Figure 7.
32
(a) original block-sparse signal (b) CoSaMP (c) model-based recovery
(RMSE = 0.723) (RMSE = 0.015)
Fig. 6. Example performance of model-based signal recovery for a block-sparse signal. (a) Example block-
compressible signal of lengthN = 4096 with K = 6 nonzero blocks of sizeJ = 64. Recovered signal from
M = 960 measurements using (b) conventional CoSaMP recovery and (c) block-based recovery.
In this case, theℓ2-norms of the blocks decay according to a power law, as described
above. Again, the number of measurements is far below the minimum number required to
guarantee stable recovery through conventional CS recovery. However, enforcing the model in the
approximation process results in a solution that is very close to the best 5-block approximation
of the signal.
Figure 8 indicates the decay in recovery error as a function of the numbers of measurements
for CoSaMP and model-based recovery. We generated sample block-sparse signals as follows:
we randomly selected a set ofK blocks, each of sizeJ , and endow them with coefficients that
follow an i.i.d. Gaussian distribution. Each sample point in the curves is generated by performing
200 trials of the corresponding algorithm. As in the connected wavelet-tree case, we observe
clear gains using model-based recovery, particularly for low-measurement regimes; CoSaMP
matches model-based recovery only forM ≥ 5K.
VII. CONCLUSIONS
In this paper, we have aimed to demonstrate that there are significant performance gains
to be made by exploiting more realistic and richer signal models beyond the simplistic sparse
and compressible models that dominate the CS literature. Building on the unions of subspaces
results of [5] and the proof machinery of [10], we have taken some of the first steps towards
33
(a) signal (b) best 5-block approximation
(RMSE = 0.116)
(c) CoSaMP (d) model-based recovery
(RMSE = 0.711) (RMSE = 0.195)
Fig. 7. Example performance of model-based signal recovery for block-compressible signals. (a) Example block-
compressible signal, lengthN = 1024. (b) Best block-based approximation withK = 5 blocks. Recovered signal
from M = 200 measurements using both (c) conventional CoSaMP recovery and (d) block-based recovery.
what promises to be a general theory for model-based CS by introducing the notion of a model-
compressible signal and the associated restricted amplification property (RAmP) condition it
imposes on the measurement matrixΦ.
For the volumes of natural and manmade signals and images that are wavelet-sparse
or compressible, our tree-based CoSaMP and IHT algorithms offer performance that signif-
icantly exceeds today’s state-of-the-art while requiringonly M = O (K) rather thanM =
O (K log(N/K)) random measurements. For block-sparse signals and signal ensembles, our
block-based CoSaMP and IHT algorithms offer not only excellent performance but also require
just M = O (JK) measurements, whereJK is the signal sparsity. Furthermore, block-based
recovery can recovery signal ensembles using fewer measurements than the number required
34
1 2 3 4 50
2
4
6
8
10
12
14
16
M/KA
vera
ge n
orm
aliz
ed e
rror
mag
nitu
de Model−based recoveryCoSaMP
Fig. 8. Performance of CoSaMP and block-based recovery on a class ofblock-sparse signals as a function ofM/K.
Standard CS recovery does not match the performance of block-based recovery untilM = 5K.
when each signal is recovered independently.
There are many avenues for future work on model-based CS. We have only considered the
recovery of signals from models that can be geometrically described as a union of subspaces;
possible extensions include other, more complex geometries (for example, high-dimensional
polytopes, nonlinear manifolds.) We also expect that the core of our proposed algorithms — a
model-enforcing approximation step — can be integrated into other iterative algorithms, such
as relaxedℓ1-norm minimization methods. Furthermore, our framework will benefit from the
formulation of new signal models that are endowed with efficient model-based approximation
algorithms.
APPENDIX I
PROOF OFTHEOREM 2
To prove this theorem, we will study the distribution of the maximum singular value of a
submatrixΦT of a matrix with i.i.d. Gaussian entriesΦ corresponding to the columns indexed
by T . From this we obtain the probability that RAmP does not hold for a fixed supportT . We
will then evaluate the same probability for all supportsT of elements ofRj,K , where the desired
bound on the amplification is dependent on the value ofj. This gives us the probability that the
35
RAmP does not hold for a given residual subspace setRj,K . We fix the probability of failure
on each of these sets; we then obtain probability that the matrix Φ does not have the RAmP
using a union bound. We end by obtaining conditions on the number of rowsM of Φ to obtain
a desired probability of failure.
We begin from the following concentration of measure for thelargest singular value of a
M ×K submatrixΦT , |T | = K, of anM ×N matrix Φ with i.i.d. subgaussian entries that are
properly normalized [18, 36, 37]:
P
(σmax(ΦT ) > 1 +
√K
M+ τ + β
)≤ e−Mτ2/2.
For large enoughM , β ≪ 1; thus we ignore this small constant in the sequel. By letting
τ = jr√
1 + ǫK − 1−√
KM
(with the appropriate value ofj for T ), we obtain
P(σmax(ΦT ) > jr
√1 + ǫK
)≤ e
−M2
“
jr√
1+ǫK−1−√
KM
”2
.
We use a union bound over all possibleRj supports foru ∈ Rj,K to obtain the probability that
Φ does not amplify the norm ofu by more thanjr√
1 + ǫK :
P(‖Φu‖2 >
(jr√
1 + ǫK)‖u‖2 ∀ u ∈ Rj,K
)≤ Rje
− 12(
√M(jr
√1+ǫK−1)−
√K)
2
.
Bound the right hand side by a constantµ; this requires
Rj ≤ e12(
√M(jr
√1+ǫK−1)−
√K)
2
µ (20)
for eachj. We use another union bound among the residual subspacesRj.K to measure the
probability that the RAmP does not hold:
P(‖Φu‖2 >
(jr√
1 + ǫK)‖u‖2 ∀ u ∈ Rj,K , ∀ j, 1 ≤ j ≤ ⌈N/K⌉
)≤⌈N
K
⌉µ.
To bound this probability bye−t, we needµ = KNe−t; plugging this into (20), we obtain
Rj ≤ e12(
√M(jr
√1+ǫK−1)−
√K)
2K
Ne−t
for eachj. Simplifying, we obtain that forΦ to posess the RAmP with probability1− e−t, the
following must hold for allj:
M ≥ 1(jr√
1 + ǫK − 1)2
(√
2
(lnRjN
K+ t
)+√K
)2
. (21)
36
Since (√a +√b)2 ≤ 2a + 2b for a, b > 0, then the hypothesis (12) implies (21), proving the
theorem.
APPENDIX II
PROOF OFTHEOREM 5
In this proof, we denoteM(x,K) = xK for brevity. To bound‖Φ(x−xK )‖2, we writex as
x = xK +
⌈N/K⌉∑
j=2
xTj,
where
xTj= xjK − x(j−1)K , j = 2, . . . , ⌈N/K⌉
is the difference between the bestjK model approximation and the best(j − 1)K model
approximation. Additionally, each piecexTj∈ Rj,K . Therefore, sinceΦ satisifes the(ǫK , s− 1)
RAmP, we obtain
‖Φ(x− xK)‖2 =
∥∥∥∥∥∥Φ
⌈N/K⌉∑
j=2
xTj
∥∥∥∥∥∥2
≤⌈N/K⌉∑
j=2
‖ΦxTj‖2 ≤
⌈N/K⌉∑
j=2
√1 + ǫKj
s−1‖xTj‖2. (22)
Sincex ∈Ms, the norm of each piece can be bounded as
‖xTj‖2 = ‖xjK − x(j−1)K‖2 ≤ ‖x− x(j−1)K‖2 + ‖x− xjK‖2 ≤ |x|MsK
−s((j − 1)−s + j−s
).
Applying this bound in (22), we obtain
‖Φ(x− xK)‖2 ≤√
1 + ǫK
⌈N/K⌉∑
j=2
js−1‖xTj‖2,
≤√
1 + ǫK |x|MsK−s
⌈N/K⌉∑
j=2
js−1((j − 1)−s + j−s),
≤√
1 + ǫK |x|MsK−s
⌈N/K⌉∑
j=2
j−1.
It is easy to show, using Euler-Maclaurin summations, that∑⌈N/K⌉
j=2 j−1 ≤ ln⌈N/K⌉; we then
obtain
‖Φ(x− xK)‖2 ≤√
1 + ǫKK−s ln
⌈N
K
⌉|x|Ms ,
which proves the theorem.
37
Algorithm 2 Model-Based Iterative Hard ThresholdingInputs: CS MatrixΦ, measurementsy, modelMK
Outpurs:K-sparse approximationx
initialize: x0 = 0, r = y, i = 0.
while halting criterion falsedo
i← i+ 1
b← xi−1 + ΦT r form signal estimatexi ←M(b,K) prune signal estimate according to signal modelr ← y − Φxi update measurement residual
end while
return x← xi
APPENDIX III
MODEL-BASED ITERATIVE HARD THRESHOLDING
Our proposed model-based iterative hard thresholding (IHT) is given in Algorithm 2. For
this algorithm, Theorems 4, 5, and 6 can be proven with only a few modifications:Φ must have
theM3K-RIP with δM3
K≤ 0.1, and the constant factor in the bound changes from 15 to 4 in
Theorem 4, from 35 to 10 in Theorem 5, and from 20 to 5 in Theorem6.
To illustrate the performance of the algorithm, we repeat the HeaviSineexperiment from
Figure 1. Recall thatN = 1024, and M = 80 for this example. The advantages of using
our tree-model-based approximation step (instead of mere hard thresholding) are evident from
Figure 9. In practice, we have observed that our model-basedalgorithm converges in fewer steps
than IHT and yields much more accurate results in terms of recovery error.
APPENDIX IV
PROOF OFTHEOREM 4
The proof of this theorem is identical to that of the CoSaMP algorithm in [10, Section 4.6],
and requires a set of six lemmas. The sequence of Lemmas 1–6 below are modifications of
38
(a) original (b) IHT (c) model-based IHT
(RMSE = 0.627) (RMSE = 0.080)
Fig. 9. Example performance of model-based IHT. (a) Piecewise-smooth HeaviSinetest signal, lengthN = 1024.
Signal recovered fromM = 80 measurements using both (b) standard and (c) model-based IHT recovery. Root mean-
squared error (RMSE) values are normalized with respect to theℓ2 norm of the signal.
the lemmas in [10] that are restricted to the signal model. Lemma 4 does not need any changes
from [10], so we state it without proof. The proof of Lemmas 3–6 use the properties in Lemmas 1
and 2, which are simple to prove.
Lemma 1:SupposeΦ hasM-RIP with constantδM. Let Ω be a support corresponding to
a subspace inM. Then we have the following handy bounds.
‖ΦTΩu‖2 ≤
√1 + δM‖u‖2,
‖Φ†Ωu‖2 ≤
1√1− δM
‖u‖2,
‖ΦTΩΦΩu‖2 ≤ (1 + δM)‖u‖2,
‖ΦTΩΦΩu‖2 ≥ (1− δM)‖u‖2,
‖(ΦTΩΦΩ)−1u‖2 ≤
1
1 + δM‖u‖2,
‖(ΦTΩΦΩ)−1u‖2 ≥
1
1− δM‖u‖2.
Lemma 2:SupposeΦ hasM2K-RIP with constantδM2
K. Let Ω be a support corresponding
to a subspace inMK , and letx ∈ MK . Then‖ΦTΩΦx|ΩC‖2 ≤ δM2
K‖x|ΩC‖2.
39
We begin the proof of Theorem 4 by fixing an iterationi ≥ 1 of model-based CoSaMP. We
write x = xi−1 for the signal estimate at the beginning of theith iteration. Define the signal
residuals = x − x, which implies thats ∈ M2K . We note that we can writer = y − Φx =
Φ(x− x) + n = Φs+ n.
Lemma 3: (Identification)The setΩ = supp(M2(e,K)), where e = ΦT r, identifies a
subspace inM2K , and obeys
‖s|ΩC‖2 ≤ 0.2223‖s‖2 + 2.34‖n‖2.
Proof of Lemma 3:Define the setΠ = supp(s). Let eΩ = M2(e,K) be the model-based
approximation toe with supportΩ, and similarly leteΠ be the approximation toe with support
Π. Each approximation is equal toe for the coefficients in the support, and zero elsewhere. Since
Ω is the support of the best approximation inM2K , we must have:
‖e− eΩ‖22 ≤ ‖e− eΠ‖22,N∑
n=1
(e[n]− eΩ[n])2 ≤N∑
n=1
(e[n]− eΠ[n])2,
∑
n/∈Ω
e[n]2 ≤∑
n/∈Π
e[n]2,
N∑
n=1
e[n]2 −∑
n/∈Ω
e[n]2 ≥N∑
n=1
e[n]2 −∑
n/∈Π
e[n]2,
∑
n∈Ω
e[n]2 ≥∑
n∈Π
e[n]2,
∑
n∈Ω
e[n]2 ≥∑
n∈Π
e[n]2,
∑
n∈Ω\Πe[n]2 ≥
∑
n∈Π\Ωe[n]2,
‖e|Ω\Π‖22 ≥ ‖e|Π\Ω‖22,
whereΩ \ Π denotes the set difference ofΩ andΠ. These signals are inM4K (since they arise
as the difference of two elements fromM2K); therefore, we can apply theM4
K-RIP constants
40
and Lemmas 1 and 2 to provide the following bounds on both sides (see [10] for details):
‖e|Ω\Π‖2 ≤ δM4K‖s‖2 +
√1 + δM2
K‖e‖2, (23)
‖e|Π\Ω‖2 ≥ (1− δM2K)‖s|ΩC‖2 − δM2
K‖s‖2 −
√1 + δM2
K‖e‖2. (24)
Combining (23) and (24), we obtain
‖s|ΩC‖2 ≤(δM2
K+ δM4
K)‖s‖2 + 2
√1 + δM2
K‖e‖2
1− δM2K
.
The argument is completed by noting thatδM2K≤ δM4
K≤ 0.1.
Lemma 4: (Support Merger)Let Ω be a set of at most2K indices. Then the setΛ =
Ω ∪ supp(x) contains at most3K indices, and‖x|ΛC‖2 ≤ ‖s|ΩC‖2.
Lemma 5: (Estimation)Let Λ be a support corresponding to a subspace inM3K , and define
the least squares signal estimateb by b|T = Φ†Ty, b|T C = 0. Then
‖x− b‖2 ≤ 1.112‖x|ΛC‖2 + 1.06‖n‖2.
Proof of Lemma 5:It can be shown [10] that
‖x− b‖2 ≤ ‖x|ΛC‖2 + ‖(ΦTΛΦΛ)−1ΦT
ΛΦx|ΠC‖2 + ‖Φ†Πn‖2.
SinceΛ is a support corresponding to a subspace inM3K and x ∈ MK , we use Lemmas 1
and 2 to obtain
‖x− b‖2 ≤ ‖x|ΛC‖2 +1
1− δM3K
‖ΦTΛΦx|ΠC‖2 +
1√1− δM3
K
‖n‖2,
≤(
1 +δM4
K
1− δM3K
)‖x|ΠC‖2 +
1√1− δM3
K
‖n‖2.
Finally, note thatδM3K≤ δM4
K≤ 0.1.
Lemma 6: (Pruning)The pruned approximationxi = M(b,K) is such that
‖x− xi‖2 ≤ 2‖x− b‖2.
41
Proof of Lemma 6: Sincexi is the best approximation inMK to b, andx ∈MK , we obtain
‖x− xi‖2 ≤ ‖x− b‖2 + ‖b− xi‖2 ≤ 2‖x− b‖2.
We use these lemmas in reverse sequence for the inequalitiesbelow:
‖x− xi‖2 ≤ 2‖x− b‖2,
≤ 2(1.112‖x|ΛC‖2 + 1.06‖n‖2),
≤ 2.224‖s|ΩC‖2 + 2.12‖n‖2,
≤ 2.224(0.2223‖s‖2 + 2.34‖n‖2) + 2.12‖n‖2,
≤ 0.5‖s‖2 + 7.5‖n‖2,
≤ 0.5‖x− xi−1‖2 + 7.5‖n‖2.
From the recursion onxi, we obtain‖x− xi‖2 ≤ 2−i‖x‖2 + 15‖n‖2. This completes the proof
of Theorem 4.
APPENDIX V
PROOF OFPROPOSITION1
WhenK < log2N , the number of subtrees of sizeK of a binary tree of sizeN is the
Catalan number [38]
TK,N =1
K + 1
(2K
K
)≤ (2e)K
K + 1,
using Stirling’s approximation. WhenK > log2N , we partition this count of subtrees into the
numbers of subtreestK,h of sizeK and heighth, to obtain
TK,N =
log2 N∑
h=⌊log2 K⌋+1
tK,h
42
We obtain the following asymptotic identity from [38, page 51]:
tK,h =4K+1.5
h4
∑
m≥1
[2K
h2(2πm)4 − 3(2πm)2
]e−
K(2πm)2
h2 + 4KO(e− ln2 h
)
+4KO(
ln8 h
h5
)+ 4KO
(ln8 h
h4
),
≤ 4K+2
h4
∑
m≥1
[2K
h2(2πm)4 − 3(2πm)2
]e−
K(2πm)2
h2 . (25)
We now simplify the formula slightly: we seek a bound for the sum term (which we denote
by βh for brevity):
βh =∑
m≥1
[2K
h2(2πm)4 − 3(2πm)2
]e−
K(2πm)2
h2 ≤∑
m≥1
2K
h2(2πm)4e−
K(2πm)2
h2 . (26)
Let mmax = hπ√
2K, the value ofm for which the term inside the sum (26) is maximum; this is
not necessarily an integer. Then,
βh ≤⌊mmax⌋−1∑
m=1
2K
h2(2πm)4e−
K(2πm)2
h2 +
⌈mmax⌉∑
m=⌊mmax⌋
2K
h2(2πm)4e−
K(2πm)2
h2
+∑
m≥⌈mmax⌉+1
2K
h2(2πm)4e−
K(2πm)2
h2 ,
≤∫ ⌊mmax⌋
1
2K
h2(2πx)4e−
K(2πx)2
h2 dx+
⌈mmax⌉∑
m=⌊mmax⌋
2K
h2(2πm)4e−
K(2πm)2
h2
+
∫ ∞
⌈mmax⌉
2K
h2(2πx)4e−
K(2πx)2
h2 dx,
where the second inequality comes from the fact that the series in the sum is strictly increasing
for m ≤ ⌊mmax⌋ and strictly decreasing form > ⌈mmax⌉. One of the terms in the sum can be
added to one of the integrals. If we have that
(2π ⌊mmax⌋)4e−K(2π⌊mmax⌋)2
h2 < (2π ⌈mmax⌉)4e−K(2π(⌈mmax⌉))2
h2 , (27)
then we can obtain
βh ≤∫ ⌈mmax⌉
1
2K
h2(2πx)4e−
K(2πx)2
h2 dx+2K
h2(2π ⌈mmax⌉)4e−
K(2π⌈mmax⌉)2
h2
+
∫ ∞
⌈mmax⌉
2K
h2(2πx)4e−
K(2πx)2
h2 dx.
43
When the opposite of (27) is true, we have that
βh ≤∫ ⌊mmax⌋
1
2K
h2(2πx)4e−
K(2πx)2
h2 dx+2K
h2(2π ⌊mmax⌋)4e−
K(2π⌊mmax⌋)2
h2
+
∫ ∞
⌊mmax⌋
2K
h2(2πx)4e−
K(2πx)2
h2 dx.
Since the term in the sum reaches its maximum formmax, we will have in all three cases that
βh ≤∫ ∞
1
2K
h2(2πx)4e−
K(2πx)2
h2 dx+8h2
Ke2.
We perform a change of variablesu = 2πx and defineσ = h/√
2K to obtain
βh ≤1
2π
∫ ∞
0
1
σ2u4e−u2/2σ2
dx+8h2
Ke2≤ 1
2σ√
2π
∫ ∞
−∞
1√2πσ
u4e−u2/2σ2
dx+8h2
Ke2.
Using the formula for the fourth central moment of a Gaussiandistribution:∫ ∞
−∞
1√2πσ
u4e−u2/2σ2
dx = 3σ4,
we obtain
βh ≤3σ3
2√
2π+
8h2
Ke2=
3h3
8√πK3
+8h2
Ke2.
Thus, (25) simplifies to
tK,h ≤4K
K
(6
h√πK
+128
h2e2
).
Correspondingly,TK,N becomes
TK,N ≤log2 N∑
h=⌊log2 K⌋+1
4K
K
(6
h√πK
+128
h2e2
),
≤ 4K
K
6√πK
log2 N∑
h=⌊log2 K⌋+1
1
h+
128
e2
log2 N∑
h=⌊log2 K⌋+1
128
h2e2
.
It is easy to show, using Euler-Maclaurin summations, that
b∑
j=a
j−1 ≤ lnb
a− 1and
b∑
j=a
j−1 ≤ 1
a− 1;
we then obtain
TK,N ≤ 4K
K
(6√πK
lnlog2N
⌊log2K⌋+
128
e2⌊log2K⌋
)≤ 4K+4
Ke2⌊log2K⌋≤ 4K+4
Ke2.
This proves the proposition.
44
APPENDIX VI
PROOF OFPROPOSITION3
We wish to find the value of the bound (12) for the subspace count given in (17). We obtain
M ≥ max1≤j≤⌈N/K⌉Mj , whereMj follows one of these three regimes:
Mj =
1
(jr√
1+ǫK−1)2
(2K + 4 ln (2e)K(2j+1)N
K(Kj+1)(Kj+K+1)+ 2t
)if j <
⌊log2 N
K
⌋,
1
(jr√
1+ǫK−1)2
(2K + 4 ln 2(3j+2)K+8ejKN
K(Kj+1)(Kj+K+1)e2 + 2t)
if j =⌊
log2 NK
⌋,
1
(jr√
1+ǫK−1)2
(2K + 4 ln 4(2j+1)K+8N
K3j(j+1)e4 + 2t)
if j >⌊
log2 NK
⌋.
We separate the terms that are linear onK and j, and obtain
Mj =
1
(jr√
1+ǫK−1)2
(K(3 + 4 ln 2) + 8Kj(1 + ln 2) + 4 ln N
K(Kj+1)(Kj+K+1) + 2t)
if j <⌊
log2
N
K
⌋,
1
(jr√
1+ǫK−1)2
(2K(1 + 4 ln 2) + 4Kj(1 + ln 8) + 4 ln 256N
K(Kj+1)(Kj+K+1)e2 + 2t)
if j =⌊
log2
N
K
⌋,
1
(jr√
1+ǫK−1)2
(2K(1 + 4 ln 2) + 16Kj ln 2 + 4 ln 65536N
K3j(j+1)e4 + 2t)
if j >⌊
log2
N
K
⌋,
=
1
(js−0.5√
1+ǫK−j−0.5)2
(8K(1 + ln 2) + K(3+4 ln 2)
j+ 4
jln N
K(Kj+1)(Kj+K+1) + 2tj
)if j <
⌊log
2N
K
⌋,
1
(js−0.5√
1+ǫK−j−0.5)2
(4K(1 + ln 8) + 2K(1+4 ln 2)
j+ 4
jln 256N
K(Kj+1)(Kj+K+1)e2 + 2tj
)if j =
⌊log
2N
K
⌋,
1
(js−0.5√
1+ǫK−j−0.5)2
(16K ln 2 + 2K(1+4 ln 2)
j+ 4
jln 65536N
K3j(j+1)e4 + 2tj
)if j >
⌊log
2N
K
⌋.
The sequencesMj⌊log2 N
K ⌋−1
j=1 and Mj⌈NK ⌉
j=⌊ log2 NK ⌋+1
are decreasing sequences, since the
numerators are decreasing sequences and the denominator isan increasing sequence whenever
s > 0.5. WhenK ≤ log2N , we have
M ≥ max
(1
(√1 + ǫK − 1
)2(K(11 + 12 ln 2) + 4 ln
N
K(K + 1)(2K + 1)+ 2t
),
4K(1 + ln 8) +2K(1+4 ln 2)+4 ln 256N
K(log2 N+1)(log2 N+K+1)e2+2t
log2 NK(
log2 NK
s−0.5√1 + ǫK − log2 N
K
−0.5)2 ,
16K ln 2 +2K(1+4 ln 2)+4 ln 65536N
K(log2 N+K)(log2 N+2K)e4+2t
log2 NK
+1((
log2 NK
+ 1)s−0.5√
1 + ǫK −(
log2 NK
+ 1)−0.5
)2
.
45
These three terms have sequentially smaller numerators andsequentially larger denominators,
resulting in
M ≥ 1(√
1 + ǫK − 1)2(K(11 + 12 ln 2) + 4 ln
N
K(K + 1)(2K + 1)+ 2t
).
WhenK > log2N , the first two regimes ofMj are nonexistent, and so we have
M ≥ 1(√
1 + ǫK − 1)2(
2K(1 + 12 ln 2) + 4 ln32768N
K3e4+ 2t
).
This completes the proof of Proposition 3.
ACKNOWLEDGEMENTS
We thank Petros Boufounos and Mark Davenport for helpful discussions.
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